Almost cyclic coherent components of an Auslander-Reiten quiver
type: Item
instance of: scholarly article
title: Almost cyclic coherent components of an Auslander-Reiten quiver
zbMATH Open document ID: 0985.16012
author: Piotr Malicki
author: Andrzej Skowroński
published in: Journal of Algebra
publication date: 2002-02-20T00:00:00Z
review text: The authors study very closely the cyclic part \(_c\Gamma_A\) of the A-R quiver of an Artin algebra \(A\). The definition of the quiver \(_c\Gamma_A\) is the following: A vertex \(X\) of the A-R quiver is called cyclic if it lies on an oriented cycle of \(\Gamma_A\). The cyclic part \(_c\Gamma_A\), of the A-R quiver, is the full translation subquiver of \(\Gamma_A\) whose vertices are the cyclic vertices. The connected components of the cyclic part are called the cyclic components of \(\Gamma_A\). One of the theorems states that two vertices are in the same cyclic component if and only if they lie on a common oriented cycle. The authors are interested in studying the cyclic coherent components. Recall that a translation subquiver \(\Gamma\) of \(\Gamma_A\) is called coherent if it satisfies the following two conditions: (C1) For each projective \(P\) in \(\Gamma\) there is an infinite sectional path \(P=X_1\to X_2\to\cdots\to X_i\to X_{i+1}\to X_{i+2}\to\cdots\) in \(\Gamma\), called a coray ending in \(P\). (C2) A similar condition has to be satisfied for injective vertices. A full translation subquiver \(\Gamma\) of \(\Gamma_A\) is said to be almost cyclic if all but a finite number of vertices of \(\Gamma\) lie on oriented cycles contained entirely in \(\Gamma\). These concepts are deeply related with the notion of coil, introduced by Assem and Skowroński, which was generalized to the notion generalized coil, by Skowroński, and now they introduce in this paper the notion of generalized multicoil. These notions are all defined in terms of a general translation quiver. I state now the most important theorems of the paper, (stated in the introduction of the paper): Theorem A. Let \(\Gamma\) be a connected component of \(\Gamma_A\). Then \(\Gamma\) is coherent and almost cyclic if and only if \(\Gamma\) is a generalized multicoil. Theorem B gets, as conclusion of Theorem A, the local structure of a coherent and almost cyclic connected component, (i.e. a generalized m…
Mathematics Subject Classification ID: 16G70
Mathematics Subject Classification ID: 16G10
Mathematics Subject Classification ID: 16G60
zbMATH DE Number: 1502172
zbMATH Keywords: Auslander-Reiten quivers
zbMATH Keywords: cyclic vertices
zbMATH Keywords: generalized multicoils
zbMATH Keywords: coherent components
zbMATH Keywords: Artin algebras
zbMATH Keywords: projective modules
zbMATH Keywords: injective modules
zbMATH Keywords: translation quivers
reviewed by: Eduardo N. Marcos
MaRDI profile type: Publication
full work available at URL: jabr.2000.8314
OpenAlex ID: W2074883738
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DOI: 10.1006/JABR.2000.8314
Wikidata QID: Q131317185
description: scientific article; zbMATH DE number 1502172